html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Cosh

 http://functions.wolfram.com/01.20.21.1442.01

 Input Form

 Integrate[z^n E^(p z^2) Cos[b z]^m Cosh[c z^2], z] == 2^(-2 - m) (z^(1 + n) Binomial[m, m/2] (((c - p) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (c - p) z^2] + ((-(c + p)) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (-(c + p)) z^2]) (Mod[m, 2] - 1) - Sum[Binomial[m, s] ((E^((b^2 (m - 2 s)^2)/(4 (-c + p))) Sqrt[c + p] Sum[(-c + p)^(-(1/2) - n) (I b (-(m/2) + s))^(n - q) ((b (m - 2 s) + 2 I (c - p) z)^2/(-c + p))^((1/2) (-1 - q)) (I b (m - 2 s) + 2 (-c + p) z)^(1 + q) Binomial[n, q] Gamma[(1 + q)/2, (b (m - 2 s) + 2 I (c - p) z)^2/(4 (-c + p))], {q, 0, n}] + E^((b^2 (m - 2 s)^2)/(4 (-c + p))) Sqrt[c + p] Sum[2^(-n + q) (-c + p)^(-(1/2) - n) (I b (m - 2 s))^(n - q) ((b (m - 2 s) + 2 I (-c + p) z)^2/(-c + p))^((1/2) (-1 - q)) ((-I) b (m - 2 s) + 2 (-c + p) z)^(1 + q) Binomial[n, q] Gamma[(1 + q)/2, (b (m - 2 s) + 2 I (-c + p) z)^2/(4 (-c + p))], {q, 0, n}] + E^((b^2 (m - 2 s)^2)/(4 (c + p))) Sqrt[-c + p] (Sum[(c + p)^(-(1/2) - n) (I b (-(m/2) + s))^(n - q) ((b (m - 2 s) - 2 I (c + p) z)^2/(c + p))^((1/2) (-1 - q)) (I b (m - 2 s) + 2 (c + p) z)^(1 + q) Binomial[n, q] Gamma[(1 + q)/2, (b (m - 2 s) - 2 I (c + p) z)^2/(4 (c + p))], {q, 0, n}] + Sum[2^(-n + q) (c + p)^(-(1/2) - n) (I b (m - 2 s))^(n - q) ((b (m - 2 s) + 2 I (c + p) z)^2/(c + p))^ ((1/2) (-1 - q)) ((-I) b (m - 2 s) + 2 (c + p) z)^(1 + q) Binomial[n, q] Gamma[(1 + q)/2, (b (m - 2 s) + 2 I (c + p) z)^2/ (4 (c + p))], {q, 0, n}]))/(Sqrt[-c + p] Sqrt[c + p])), {s, 0, Floor[(1/2) (-1 + m)]}]) /; Element[m, Integers] && m > 0 && Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", SuperscriptBox["z", "2"]]]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["b", " ", "z"]], "]"]], "m"], " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", SuperscriptBox["z", "2"]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "m"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List["1", "+", "n"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["c", "-", "p"]], ")"]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "n"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "n"]], "2"], ",", RowBox[List[RowBox[List["(", RowBox[List["c", "-", "p"]], ")"]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]]]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "n"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "n"]], "2"], ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]], "-", "1"]], ")"]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "s"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], "2"]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "p"]], ")"]]]]]], " ", SqrtBox[RowBox[List["c", "+", "p"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "p"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["m", "2"]]], "+", "s"]], ")"]]]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "-", "p"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List[RowBox[List["-", "c"]], "+", "p"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "p"]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "-", "p"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "p"]], ")"]]]]]]], "]"]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], "2"]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "p"]], ")"]]]]]], " ", SqrtBox[RowBox[List["c", "+", "p"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "p"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "p"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List[RowBox[List["-", "c"]], "+", "p"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "p"]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "p"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "p"]], ")"]]]]]]], "]"]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], "2"]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]]]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "c"]], "+", "p"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["m", "2"]]], "+", "s"]], ")"]]]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List["c", "+", "p"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]]]]]]], "]"]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List["c", "+", "p"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]]]]]]], "]"]]]]]]]], ")"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "c"]], "+", "p"]]], " ", SqrtBox[RowBox[List["c", "+", "p"]]]]], ")"]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 z n p z 2 cos m ( b z ) cosh ( c z 2 ) z 2 - m - 2 ( z n + 1 ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox[FractionBox["m", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( Γ ( n + 1 2 , ( c - p ) z 2 ) ( ( c - p ) z 2 ) 1 2 ( - n - 1 ) + ( - ( c + p ) z 2 ) 1 2 ( - n - 1 ) Γ ( n + 1 2 , - ( c + p ) z 2 ) ) ( m mod 2 \$CellContext`m 2 - 1 ) - s = 0 m - 1 2 1 p - c c + p ( ( m s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( b 2 ( m - 2 s ) 2 4 ( p - c ) c + p q = 0 n ( p - c ) - n - 1 2 ( b ( s - m 2 ) ) n - q ( ( b ( m - 2 s ) + 2 ( c - p ) z ) 2 p - c ) 1 2 ( - q - 1 ) ( b ( m - 2 s ) + 2 ( p - c ) z ) q + 1 ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( b ( m - 2 s ) + 2 ( c - p ) z ) 2 4 ( p - c ) ) + b 2 ( m - 2 s ) 2 4 ( p - c ) c + p q = 0 n 2 q - n ( p - c ) - n - 1 2 ( b ( m - 2 s ) ) n - q ( ( b ( m - 2 s ) + 2 ( p - c ) z ) 2 p - c ) 1 2 ( - q - 1 ) ( 2 ( p - c ) z - b ( m - 2 s ) ) q + 1 ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( b ( m - 2 s ) + 2 ( p - c ) z ) 2 4 ( p - c ) ) + b 2 ( m - 2 s ) 2 4 ( c + p ) p - c ( q = 0 n 2 q - n ( c + p ) - n - 1 2 ( b ( m - 2 s ) ) n - q ( ( b ( m - 2 s ) + 2 ( c + p ) z ) 2 c + p ) 1 2 ( - q - 1 ) ( 2 ( c + p ) z - b ( m - 2 s ) ) q + 1 ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( b ( m - 2 s ) + 2 ( c + p ) z ) 2 4 ( c + p ) ) + q = 0 n ( c + p ) - n - 1 2 ( b ( s - m 2 ) ) n - q ( ( b ( m - 2 s ) - 2 ( c + p ) z ) 2 c + p ) 1 2 ( - q - 1 ) ( b ( m - 2 s ) + 2 ( c + p ) z ) q + 1 ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( b ( m - 2 s ) - 2 ( c + p ) z ) 2 4 ( c + p ) ) ) ) ) ) /; m TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] m > 0 n Condition z z n p z 2 b z m c z 2 2 -1 m -2 z n 1 Binomial m m 2 -1 Gamma n 1 2 -1 c -1 p z 2 c -1 p z 2 1 2 -1 n -1 -1 c p z 2 1 2 -1 n -1 Gamma n 1 2 -1 -1 c p z 2 \$CellContext`m 2 -1 -1 s 0 m -1 2 -1 1 p -1 c 1 2 c p 1 2 -1 Binomial m s b 2 m -1 2 s 2 4 p -1 c -1 c p 1 2 q 0 n p -1 c -1 n -1 1 2 b s -1 m 2 -1 n -1 q b m -1 2 s 2 c -1 p z 2 p -1 c -1 1 2 -1 q -1 b m -1 2 s 2 p -1 c z q 1 Binomial n q Gamma q 1 2 -1 b m -1 2 s 2 c -1 p z 2 4 p -1 c -1 b 2 m -1 2 s 2 4 p -1 c -1 c p 1 2 q 0 n 2 q -1 n p -1 c -1 n -1 1 2 b m -1 2 s n -1 q b m -1 2 s 2 p -1 c z 2 p -1 c -1 1 2 -1 q -1 2 p -1 c z -1 b m -1 2 s q 1 Binomial n q Gamma q 1 2 -1 b m -1 2 s 2 p -1 c z 2 4 p -1 c -1 b 2 m -1 2 s 2 4 c p -1 p -1 c 1 2 q 0 n 2 q -1 n c p -1 n -1 1 2 b m -1 2 s n -1 q b m -1 2 s 2 c p z 2 c p -1 1 2 -1 q -1 2 c p z -1 b m -1 2 s q 1 Binomial n q Gamma q 1 2 -1 b m -1 2 s 2 c p z 2 4 c p -1 q 0 n c p -1 n -1 1 2 b s -1 m 2 -1 n -1 q b m -1 2 s -1 2 c p z 2 c p -1 1 2 -1 q -1 b m -1 2 s 2 c p z q 1 Binomial n q Gamma q 1 2 -1 b m -1 2 s -1 2 c p z 2 4 c p -1 m m 0 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "n_"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["p_", " ", SuperscriptBox["z_", "2"]]]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["b_", " ", "z_"]], "]"]], "m_"], " ", RowBox[List["Cosh", "[", RowBox[List["c_", " ", SuperscriptBox["z_", "2"]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "m"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List["1", "+", "n"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["c", "-", "p"]], ")"]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "n"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "n"]], "2"], ",", RowBox[List[RowBox[List["(", RowBox[List["c", "-", "p"]], ")"]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]]]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "n"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "n"]], "2"], ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]], "-", "1"]], ")"]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], FractionBox[RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], "2"]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "p"]], ")"]]]]]], " ", SqrtBox[RowBox[List["c", "+", "p"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "p"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["m", "2"]]], "+", "s"]], ")"]]]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "-", "p"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List[RowBox[List["-", "c"]], "+", "p"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "p"]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "-", "p"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "p"]], ")"]]]]]]], "]"]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], "2"]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "p"]], ")"]]]]]], " ", SqrtBox[RowBox[List["c", "+", "p"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "p"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "p"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List[RowBox[List["-", "c"]], "+", "p"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "p"]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "p"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "p"]], ")"]]]]]]], "]"]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], "2"]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]]]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "c"]], "+", "p"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["m", "2"]]], "+", "s"]], ")"]]]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List["c", "+", "p"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]]]]]]], "]"]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List["c", "+", "p"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]]]]]]], "]"]]]]]]]], ")"]]]]]], ")"]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "c"]], "+", "p"]]], " ", SqrtBox[RowBox[List["c", "+", "p"]]]]]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2002-12-18